=====Modular lattices=====
Abbreviation: **MLat**
====Definition====
A \emph{modular lattice} is a [[lattice]] $\mathbf{L}=\langle L, \vee, \wedge\rangle$ that satisfies the
\emph{modular identity}: $((x\wedge z) \vee y) \wedge z = (x\wedge z) \vee (y\wedge z)$
====Definition====
A \emph{modular lattice} is a [[lattice]] $\mathbf{L}=\langle L, \vee, \wedge\rangle$ that satisfies the
\emph{modular law}: $x\le z\Longrightarrow (x\vee y) \wedge z\le x\vee (y\wedge z)$
====Definition====
A \emph{modular lattice} is a lattice $\mathbf{L}=\langle L,\vee,\wedge\rangle $ such that $\mathbf{L}$ has no sublattice isomorphic
to the pentagon $\mathbf{N}_{5}$
==Morphisms==
Let $\mathbf{L}$ and $\mathbf{M}$ be modular lattices.
A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a
homomorphism:
$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$
====Examples====
Example 1: $M_3$
is the smallest nondistributive modular lattice. By a result of [(Dedekind1900)]
this lattice occurs as a sublattice of every nondistributive
modular lattice.
====Basic results====
====Properties====
^[[Classtype]] |variety |
^[[Equational theory]] |undecidable [(Freese1980)] [(Herrmann1983)] |
^[[Quasiequational theory]] |undecidable [(Lipshitz1974)] |
^[[First-order theory]] |undecidable |
^[[Locally finite]] |no |
^[[Residual size]] |unbounded |
^[[Congruence distributive]] |yes |
^[[Congruence modular]] |yes |
^[[Congruence n-permutable]] |no |
^[[Congruence regular]] |no |
^[[Congruence uniform]] |no |
^[[Congruence extension property]] |no |
^[[Definable principal congruences]] |no |
^[[Equationally def. pr. cong.]] |no |
^[[Amalgamation property]] |no |
^[[Strong amalgamation property]] |no |
^[[Epimorphisms are surjective]] |no |
====Finite members====
$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &1\\
f(4)= &2\\
f(5)= &4\\
f(6)= &8\\
f(7)= &16\\
f(8)= &34\\
f(9)= &72\\
f(10)= &157\\
f(11)= &343\\
f(12)= &766\\
f(13)= &1718\\
f(14)= &3899\\
f(15)= &8898\\
f(16)= &20475\\
f(17)= &47321\\
f(18)= &110024\\
f(19)= &256791\\
f(20)= &601991\\
f(21)= &1415768\\
f(22)= &3340847\\
f(23)= &7904700\\
f(24)= &18752942\\
f(25)= &\\
f(26)= &\\
\end{array}$[(Peter Jipsen, Nathan Lawless, \emph{Generating all finite modular lattices of a given size},
Algebra Universalis, \textbf{74}, 2015, 253--264)]
====Subclasses====
[[Distributive lattices]]
[[Complete modular lattices]]
====Superclasses====
[[Semimodular lattices]]
[[Geometric lattices]]
====References====
[(Dedekind1900>
Richard Dedekind, \emph{\"Uber die von drei Moduln erzeugte Dualgruppe},
Math. Ann., \textbf{53}, 1900, 371--403)]
[(Freese1980>
Ralph Freese, \emph{Free modular lattices},
Trans. Amer. Math. Soc., \textbf{261}, 1980, 81--91)]
[(Herrmann1983>
Christian Herrmann, \emph{On the word problem for the modular lattice with four free generators},
Math. Ann., \textbf{265}, 1983, 513--527)]
[(Lipshitz1974>
L. Lipshitz, \emph{The undecidability of the word problems for projective geometries and modular lattices},
Trans. Amer. Math. Soc., \textbf{193}, 1974, 171--180)]